Advanced Calculus Single Variable

4.2 Sequences

Functions defined on the set of integers larger than a given integer are called sequences.

Definition 4.2.1A function whosedomain is defined as a set of the form

{k,k + 1,k + 2,⋅⋅⋅}

for k an integer is known as a sequence. Thus you can consider f

(k)

,f

(k +1)

,f

(k + 2)

,etc. Usually the domain of the sequence is either ℕ, the natural numbers consisting of

{1,2,3,⋅⋅⋅}

or the nonnegative integers,

{0,1,2,3,⋅⋅⋅}

. Also, it is traditional towrite f1,f2, etc. instead of f

(1)

,f

(2)

,f

(3)

etc. when referring to sequences. Inthe above context, fkis called the first term, fk+1the second and so forth. It isalso common to write the sequence, not as f but as

{fi}

i=k∞or just

{fi}

forshort.

Example 4.2.2Let

{ak}

k=1∞be defined by ak≡ k2 + 1.

This gives a sequence. In fact, a7 = a

(7)

= 72 + 1 = 50 just from using the formula for the
kth term of the sequence.

It is nice when sequences come in this way from a formula for the kth term.
However, this is often not the case. Sometimes sequences are defined recursively. This
happens, when the first several terms of the sequence are given and then a rule is
specified which determines an+1 from knowledge of a1,

⋅⋅⋅

,an. This rule which
specifies an+1 from knowledge of ak for k ≤ n is known as a recurrence relation.

Example 4.2.3Let a1 = 1 and a2 = 1. Assuming a1,

⋅⋅⋅

,an+1are known, an+2≡an + an+1.

Thus the first several terms of this sequence, listed in order, are 1, 1, 2, 3, 5, 8,

⋅⋅⋅

. This
particular sequence is called the Fibonacci sequence and is important in the study of
reproducing rabbits. Note this defines a function without giving a formula for it. Such
sequences occur naturally in the solution of differential equations using power series methods
and in many other situations of great importance.

For sequences, it is very important to consider something called a subsequence.

Definition 4.2.4Let

{an}

be a sequence and let n1< n2< n3,

⋅⋅⋅

be anystrictly increasing list of integers such that n1is at least as large as the first number inthe domain of the function. Then if bk≡ ank,